Combining Philosophers

All the ideas for Aeschylus, E.J. Lemmon and Jos A. Benardete

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79 ideas

1. Philosophy / E. Nature of Metaphysics / 1. Nature of Metaphysics
3358 Metaphysics focuses on Platonism, essentialism, materialism and anti-realism [Benardete,JA]
     Full Idea: In contemporary metaphysics the major areas of discussion are Platonism, essentialism, materialism and anti-realism.
     From: José A. Benardete (Metaphysics: the logical approach [1989], After)
1. Philosophy / F. Analytic Philosophy / 5. Linguistic Analysis
3312 There are the 'is' of predication (a function), the 'is' of identity (equals), and the 'is' of existence (quantifier) [Benardete,JA]
     Full Idea: At least since Russell, one has routinely distinguished between the 'is' of predication ('Socrates is wise', Fx), the 'is' of identity ('Morning Star is Evening Star', =), and the 'is' of existence ('the cat is under the bed', Ex).
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch. 7)
     A reaction: This seems horribly nitpicking to many people, but I love it - because it is just true, and it is a truth right at the basis of the confusions in our talk. Analytic philosophy forever! [P.S. 'Tiddles is a cat' - the 'is' membership]
1. Philosophy / F. Analytic Philosophy / 7. Limitations of Analysis
3352 Analytical philosophy analyses separate concepts successfully, but lacks a synoptic vision of the results [Benardete,JA]
     Full Idea: Analytical philosophy excels in the piecemeal analysis of causation, perception, knowledge and so on, but there is a striking poverty of any synoptic vision of these independent studies.
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.22)
1. Philosophy / G. Scientific Philosophy / 1. Aims of Science
3329 Presumably the statements of science are true, but should they be taken literally or not? [Benardete,JA]
     Full Idea: As our bible, the Book of Science is presumed to contain only true sentences, but it is less clear how they are to be construed, which literally and which non-literally.
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.13)
4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
9535 'Contradictory' propositions always differ in truth-value [Lemmon]
     Full Idea: Two propositions are 'contradictory' if they are never both true and never both false either, which means that ¬(A↔B) is a tautology.
     From: E.J. Lemmon (Beginning Logic [1965], 2.3)
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / a. Symbols of PL
9509 That proposition that both P and Q is their 'conjunction', written P∧Q [Lemmon]
     Full Idea: If P and Q are any two propositions, the proposition that both P and Q is called the 'conjunction' of P and Q, and is written P∧Q.
     From: E.J. Lemmon (Beginning Logic [1965], 1.3)
     A reaction: [I use the more fashionable inverted-v '∧', rather than Lemmon's '&', which no longer seems to be used] P∧Q can also be defined as ¬(¬P∨¬Q)
9508 The sign |- may be read as 'therefore' [Lemmon]
     Full Idea: I introduce the sign |- to mean 'we may validly conclude'. To call it the 'assertion sign' is misleading. It may conveniently be read as 'therefore'.
     From: E.J. Lemmon (Beginning Logic [1965], 1.2)
     A reaction: [Actually no gap between the vertical and horizontal strokes of the sign] As well as meaning 'assertion', it may also mean 'it is a theorem that' (with no proof shown).
9511 We write the conditional 'if P (antecedent) then Q (consequent)' as P→Q [Lemmon]
     Full Idea: We write 'if P then Q' as P→Q. This is called a 'conditional', with P as its 'antecedent', and Q as its 'consequent'.
     From: E.J. Lemmon (Beginning Logic [1965], 1.2)
     A reaction: P→Q can also be written as ¬P∨Q.
9510 That proposition that either P or Q is their 'disjunction', written P∨Q [Lemmon]
     Full Idea: If P and Q are any two propositions, the proposition that either P or Q is called the 'disjunction' of P and Q, and is written P∨Q.
     From: E.J. Lemmon (Beginning Logic [1965], 1.3)
     A reaction: This is inclusive-or (meaning 'P, or Q, or both'), and not exlusive-or (Boolean XOR), which means 'P, or Q, but not both'. The ∨ sign is sometimes called 'vel' (Latin).
9512 We write the 'negation' of P (not-P) as ¬ [Lemmon]
     Full Idea: We write 'not-P' as ¬P. This is called the 'negation' of P. The 'double negation' of P (not not-P) would be written as ¬¬P.
     From: E.J. Lemmon (Beginning Logic [1965], 1.2)
     A reaction: Lemmons use of -P is no longer in use for 'not'. A tilde sign (squiggle) is also used for 'not', but some interpreters give that a subtly different meaning (involving vagueness). The sign ¬ is sometimes called 'hook' or 'corner'.
9513 We write 'P if and only if Q' as P↔Q; it is also P iff Q, or (P→Q)∧(Q→P) [Lemmon]
     Full Idea: We write 'P if and only if Q' as P↔Q. It is called the 'biconditional', often abbreviate in writing as 'iff'. It also says that P is both sufficient and necessary for Q, and may be written out in full as (P→Q)∧(Q→P).
     From: E.J. Lemmon (Beginning Logic [1965], 1.4)
     A reaction: If this symbol is found in a sequence, the first move in a proof is to expand it to the full version.
9514 If A and B are 'interderivable' from one another we may write A -||- B [Lemmon]
     Full Idea: If we say that A and B are 'interderivable' from one another (that is, A |- B and B |- A), then we may write A -||- B.
     From: E.J. Lemmon (Beginning Logic [1965], 1.5)
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
9516 A 'well-formed formula' follows the rules for variables, ¬, →, ∧, ∨, and ↔ [Lemmon]
     Full Idea: A 'well-formed formula' of the propositional calculus is a sequence of symbols which follows the rules for variables, ¬, →, ∧, ∨, and ↔.
     From: E.J. Lemmon (Beginning Logic [1965], 2.1)
9517 The 'scope' of a connective is the connective, the linked formulae, and the brackets [Lemmon]
     Full Idea: The 'scope' of a connective in a certain formula is the formulae linked by the connective, together with the connective itself and the (theoretically) encircling brackets
     From: E.J. Lemmon (Beginning Logic [1965], 2.1)
9519 A 'substitution-instance' is a wff formed by consistent replacing variables with wffs [Lemmon]
     Full Idea: A 'substitution-instance' is a wff which results by replacing one or more variables throughout with the same wffs (the same wff replacing each variable).
     From: E.J. Lemmon (Beginning Logic [1965], 2.2)
9529 A wff is 'inconsistent' if all assignments to variables result in the value F [Lemmon]
     Full Idea: If a well-formed formula of propositional calculus takes the value F for all possible assignments of truth-values to its variables, it is said to be 'inconsistent'.
     From: E.J. Lemmon (Beginning Logic [1965], 2.3)
9531 'Contrary' propositions are never both true, so that ¬(A∧B) is a tautology [Lemmon]
     Full Idea: If A and B are expressible in propositional calculus notation, they are 'contrary' if they are never both true, which may be tested by the truth-table for ¬(A∧B), which is a tautology if they are contrary.
     From: E.J. Lemmon (Beginning Logic [1965], 2.3)
9534 Two propositions are 'equivalent' if they mirror one another's truth-value [Lemmon]
     Full Idea: Two propositions are 'equivalent' if whenever A is true B is true, and whenever B is true A is true, in which case A↔B is a tautology.
     From: E.J. Lemmon (Beginning Logic [1965], 2.3)
9530 A wff is 'contingent' if produces at least one T and at least one F [Lemmon]
     Full Idea: If a well-formed formula of propositional calculus takes at least one T and at least one F for all the assignments of truth-values to its variables, it is said to be 'contingent'.
     From: E.J. Lemmon (Beginning Logic [1965], 2.3)
9532 'Subcontrary' propositions are never both false, so that A∨B is a tautology [Lemmon]
     Full Idea: If A and B are expressible in propositional calculus notation, they are 'subcontrary' if they are never both false, which may be tested by the truth-table for A∨B, which is a tautology if they are subcontrary.
     From: E.J. Lemmon (Beginning Logic [1965], 2.3)
9533 A 'implies' B if B is true whenever A is true (so that A→B is tautologous) [Lemmon]
     Full Idea: One proposition A 'implies' a proposition B if whenever A is true B is true (but not necessarily conversely), which is only the case if A→B is tautologous. Hence B 'is implied' by A.
     From: E.J. Lemmon (Beginning Logic [1965], 2.3)
9528 A wff is a 'tautology' if all assignments to variables result in the value T [Lemmon]
     Full Idea: If a well-formed formula of propositional calculus takes the value T for all possible assignments of truth-values to its variables, it is said to be a 'tautology'.
     From: E.J. Lemmon (Beginning Logic [1965], 2.3)
9518 A 'theorem' is the conclusion of a provable sequent with zero assumptions [Lemmon]
     Full Idea: A 'theorem' of logic is the conclusion of a provable sequent in which the number of assumptions is zero.
     From: E.J. Lemmon (Beginning Logic [1965], 2.2)
     A reaction: This is what Quine and others call a 'logical truth'.
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
9396 DN: Given A, we may derive ¬¬A [Lemmon]
     Full Idea: Double Negation (DN): Given A, we may derive ¬¬A as a conclusion, and vice versa. The conclusion depends on the assumptions of the premiss.
     From: E.J. Lemmon (Beginning Logic [1965], 1.5)
9393 A: we may assume any proposition at any stage [Lemmon]
     Full Idea: Assumptions (A): any proposition may be introduced at any stage of a proof.
     From: E.J. Lemmon (Beginning Logic [1965], 1.5)
9399 ∧E: Given A∧B, we may derive either A or B separately [Lemmon]
     Full Idea: And-Elimination (∧E): Given A∧B, we may derive either A or B separately. The conclusions will depend on the assumptions of the premiss.
     From: E.J. Lemmon (Beginning Logic [1965], 1.5)
9401 ∨E: Derive C from A∨B, if C can be derived both from A and from B [Lemmon]
     Full Idea: Or-Elimination (∨E): Given A∨B, we may derive C if it is proved from A as assumption and from B as assumption. This will also depend on prior assumptions.
     From: E.J. Lemmon (Beginning Logic [1965], 1.5)
9398 ∧I: Given A and B, we may derive A∧B [Lemmon]
     Full Idea: And-Introduction (&I): Given A and B, we may derive A∧B as conclusion. This depends on their previous assumptions.
     From: E.J. Lemmon (Beginning Logic [1965], 1.5)
9397 CP: Given a proof of B from A as assumption, we may derive A→B [Lemmon]
     Full Idea: Conditional Proof (CP): Given a proof of B from A as assumption, we may derive A→B as conclusion, on the remaining assumptions (if any).
     From: E.J. Lemmon (Beginning Logic [1965], 1.5)
9394 MPP: Given A and A→B, we may derive B [Lemmon]
     Full Idea: Modus Ponendo Ponens (MPP): Given A and A→B, we may derive B as a conclusion. B will rest on any assumptions that have been made.
     From: E.J. Lemmon (Beginning Logic [1965], 1.5)
9402 RAA: If assuming A will prove B∧¬B, then derive ¬A [Lemmon]
     Full Idea: Reduction ad Absurdum (RAA): Given a proof of B∧¬B from A as assumption, we may derive ¬A as conclusion, depending on the remaining assumptions (if any).
     From: E.J. Lemmon (Beginning Logic [1965], 1.5)
9395 MTT: Given ¬B and A→B, we derive ¬A [Lemmon]
     Full Idea: Modus Tollendo Tollens (MTT): Given ¬B and A→B, we derive ¬A as a conclusion. ¬A depends on any assumptions that have been made
     From: E.J. Lemmon (Beginning Logic [1965], 1.5)
9400 ∨I: Given either A or B separately, we may derive A∨B [Lemmon]
     Full Idea: Or-Introduction (∨I): Given either A or B separately, we may derive A∨B as conclusion. This depends on the assumption of the premisses.
     From: E.J. Lemmon (Beginning Logic [1965], 1.5)
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
9521 'Modus tollendo ponens' (MTP) says ¬P, P ∨ Q |- Q [Lemmon]
     Full Idea: 'Modus tollendo ponens' (MTP) says that if a disjunction holds and also the negation of one of its disjuncts, then the other disjunct holds. Thus ¬P, P ∨ Q |- Q may be introduced as a theorem.
     From: E.J. Lemmon (Beginning Logic [1965], 2.2)
     A reaction: Unlike Modus Ponens and Modus Tollens, this is a derived rule.
9522 'Modus ponendo tollens' (MPT) says P, ¬(P ∧ Q) |- ¬Q [Lemmon]
     Full Idea: 'Modus ponendo tollens' (MPT) says that if the negation of a conjunction holds and also one of its conjuncts, then the negation of the other conjunct holds. Thus P, ¬(P ∧ Q) |- ¬Q may be introduced as a theorem.
     From: E.J. Lemmon (Beginning Logic [1965], 2.2)
     A reaction: Unlike Modus Ponens and Modus Tollens, this is a derived rule.
9525 We can change conditionals into negated conjunctions with P→Q -||- ¬(P ∧ ¬Q) [Lemmon]
     Full Idea: The proof that P→Q -||- ¬(P ∧ ¬Q) is useful for enabling us to change conditionals into negated conjunctions
     From: E.J. Lemmon (Beginning Logic [1965], 2.2)
9524 We can change conditionals into disjunctions with P→Q -||- ¬P ∨ Q [Lemmon]
     Full Idea: The proof that P→Q -||- ¬P ∨ Q is useful for enabling us to change conditionals into disjunctions.
     From: E.J. Lemmon (Beginning Logic [1965], 2.2)
9523 De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions [Lemmon]
     Full Idea: The forms of De Morgan's Laws [P∨Q -||- ¬(¬P ∧ ¬Q); ¬(P∨Q) -||- ¬P ∧ ¬Q; ¬(P∧Q) -||- ¬P ∨ ¬Q); P∧Q -||- ¬(¬P∨¬Q)] transform negated conjunctions and disjunctions into non-negated disjunctions and conjunctions respectively.
     From: E.J. Lemmon (Beginning Logic [1965], 2.2)
9527 The Distributive Laws can rearrange a pair of conjunctions or disjunctions [Lemmon]
     Full Idea: The Distributive Laws say that P ∧ (Q∨R) -||- (P∧Q) ∨ (P∧R), and that P ∨ (Q∨R) -||- (P∨Q) ∧ (P∨R)
     From: E.J. Lemmon (Beginning Logic [1965], 2.2)
9526 We can change conjunctions into negated conditionals with P→Q -||- ¬(P → ¬Q) [Lemmon]
     Full Idea: The proof that P∧Q -||- ¬(P → ¬Q) is useful for enabling us to change conjunctions into negated conditionals.
     From: E.J. Lemmon (Beginning Logic [1965], 2.2)
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
9537 Truth-tables are good for showing invalidity [Lemmon]
     Full Idea: The truth-table approach enables us to show the invalidity of argument-patterns, as well as their validity.
     From: E.J. Lemmon (Beginning Logic [1965], 2.4)
9538 A truth-table test is entirely mechanical, but this won't work for more complex logic [Lemmon]
     Full Idea: A truth-table test is entirely mechanical, ..and in propositional logic we can even generate proofs mechanically for tautological sequences, ..but this mechanical approach breaks down with predicate calculus, and proof-discovery is an imaginative process.
     From: E.J. Lemmon (Beginning Logic [1965], 2.5)
4. Formal Logic / B. Propositional Logic PL / 4. Soundness of PL
9536 If any of the nine rules of propositional logic are applied to tautologies, the result is a tautology [Lemmon]
     Full Idea: If any application of the nine derivation rules of propositional logic is made on tautologous sequents, we have demonstrated that the result is always a tautologous sequent. Thus the system is consistent.
     From: E.J. Lemmon (Beginning Logic [1965], 2.4)
     A reaction: The term 'sound' tends to be used now, rather than 'consistent'. See Lemmon for the proofs of each of the nine rules.
4. Formal Logic / B. Propositional Logic PL / 5. Completeness of PL
9539 Propositional logic is complete, since all of its tautologous sequents are derivable [Lemmon]
     Full Idea: A logical system is complete is all expressions of a specified kind are derivable in it. If we specify tautologous sequent-expressions, then propositional logic is complete, because we can show that all tautologous sequents are derivable.
     From: E.J. Lemmon (Beginning Logic [1965], 2.5)
     A reaction: [See Lemmon 2.5 for details of the proofs]
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / a. Symbols of PC
13909 Write '(∀x)(...)' to mean 'take any x: then...', and '(∃x)(...)' to mean 'there is an x such that....' [Lemmon]
     Full Idea: Just as '(∀x)(...)' is to mean 'take any x: then....', so we write '(∃x)(...)' to mean 'there is an x such that....'
     From: E.J. Lemmon (Beginning Logic [1965], 3.1)
     A reaction: [Actually Lemmon gives the universal quantifier symbol as '(x)', but the inverted A ('∀') seems to have replaced it these days]
13902 'Gm' says m has property G, and 'Pmn' says m has relation P to n [Lemmon]
     Full Idea: A predicate letter followed by one name expresses a property ('Gm'), and a predicate-letter followed by two names expresses a relation ('Pmn'). We could write 'Pmno' for a complex relation like betweenness.
     From: E.J. Lemmon (Beginning Logic [1965], 3.1)
13911 The 'symbols' are bracket, connective, term, variable, predicate letter, reverse-E [Lemmon]
     Full Idea: I define a 'symbol' (of the predicate calculus) as either a bracket or a logical connective or a term or an individual variable or a predicate-letter or reverse-E (∃).
     From: E.J. Lemmon (Beginning Logic [1965], 4.1)
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / b. Terminology of PC
13910 Our notation uses 'predicate-letters' (for 'properties'), 'variables', 'proper names', 'connectives' and 'quantifiers' [Lemmon]
     Full Idea: Quantifier-notation might be thus: first, render into sentences about 'properties', and use 'predicate-letters' for them; second, introduce 'variables'; third, introduce propositional logic 'connectives' and 'quantifiers'. Plus letters for 'proper names'.
     From: E.J. Lemmon (Beginning Logic [1965], 3.1)
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / c. Derivations rules of PC
13904 Universal Elimination (UE) lets us infer that an object has F, from all things having F [Lemmon]
     Full Idea: Our rule of universal quantifier elimination (UE) lets us infer that any particular object has F from the premiss that all things have F. It is a natural extension of &E (and-elimination), as universal propositions generally affirm a complex conjunction.
     From: E.J. Lemmon (Beginning Logic [1965], 3.2)
13906 With finite named objects, we can generalise with &-Intro, but otherwise we need ∀-Intro [Lemmon]
     Full Idea: If there are just three objects and each has F, then by an extension of &I we are sure everything has F. This is of no avail, however, if our universe is infinitely large or if not all objects have names. We need a new device, Universal Introduction, UI.
     From: E.J. Lemmon (Beginning Logic [1965], 3.2)
13908 UE all-to-one; UI one-to-all; EI arbitrary-to-one; EE proof-to-one [Lemmon]
     Full Idea: Univ Elim UE - if everything is F, then something is F; Univ Intro UI - if an arbitrary thing is F, everything is F; Exist Intro EI - if an arbitrary thing is F, something is F; Exist Elim EE - if a proof needed an object, there is one.
     From: E.J. Lemmon (Beginning Logic [1965], 3.3)
     A reaction: [My summary of Lemmon's four main rules for predicate calculus] This is the natural deduction approach, of trying to present the logic entirely in terms of introduction and elimination rules. See Bostock on that.
13901 Predicate logic uses propositional connectives and variables, plus new introduction and elimination rules [Lemmon]
     Full Idea: In predicate calculus we take over the propositional connectives and propositional variables - but we need additional rules for handling quantifiers: four rules, an introduction and elimination rule for the universal and existential quantifiers.
     From: E.J. Lemmon (Beginning Logic [1965])
     A reaction: This is Lemmon's natural deduction approach (invented by Gentzen), which is largely built on introduction and elimination rules.
13903 Universal elimination if you start with the universal, introduction if you want to end with it [Lemmon]
     Full Idea: The elimination rule for the universal quantifier concerns the use of a universal proposition as a premiss to establish some conclusion, whilst the introduction rule concerns what is required by way of a premiss for a universal proposition as conclusion.
     From: E.J. Lemmon (Beginning Logic [1965], 3.2)
     A reaction: So if you start with the universal, you need to eliminate it, and if you start without it you need to introduce it.
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀
13905 If there is a finite domain and all objects have names, complex conjunctions can replace universal quantifiers [Lemmon]
     Full Idea: If all objects in a given universe had names which we knew and there were only finitely many of them, then we could always replace a universal proposition about that universe by a complex conjunction.
     From: E.J. Lemmon (Beginning Logic [1965], 3.2)
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃
13900 'Some Frenchmen are generous' is rendered by (∃x)(Fx→Gx), and not with the conditional → [Lemmon]
     Full Idea: It is a common mistake to render 'some Frenchmen are generous' by (∃x)(Fx→Gx) rather than the correct (∃x)(Fx&Gx). 'All Frenchmen are generous' is properly rendered by a conditional, and true if there are no Frenchmen.
     From: E.J. Lemmon (Beginning Logic [1965], 3.1)
     A reaction: The existential quantifier implies the existence of an x, but the universal quantifier does not.
4. Formal Logic / F. Set Theory ST / 1. Set Theory
3326 Set theory attempts to reduce the 'is' of predication to mathematics [Benardete,JA]
     Full Idea: Set theory offers the promise of a complete mathematization of the 'is' of predication.
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.13)
3327 The set of Greeks is included in the set of men, but isn't a member of it [Benardete,JA]
     Full Idea: Set inclusion is sharply distinguished from set membership (as the set of Greeks is found to be included in, but not a member of, the set of men).
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.13)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
3335 The standard Z-F Intuition version of set theory has about ten agreed axioms [Benardete,JA, by PG]
     Full Idea: Zermelo proposed seven axioms for set theory, with Fraenkel adding others, to produce the standard Z-F Intuition.
     From: report of José A. Benardete (Metaphysics: the logical approach [1989], Ch.17) by PG - Db (ideas)
5. Theory of Logic / B. Logical Consequence / 8. Material Implication
9520 The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q [Lemmon]
     Full Idea: The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q. That is, since Napoleon was French, then if the moon is blue then Napoleon was French; and since Napoleon was not Chinese, then if Napoleon was Chinese, the moon is blue.
     From: E.J. Lemmon (Beginning Logic [1965], 2.2)
     A reaction: This is why the symbol → does not really mean the 'if...then' of ordinary English. Russell named it 'material implication' to show that it was a distinctively logical operator.
6. Mathematics / A. Nature of Mathematics / 2. Geometry
3332 Greeks saw the science of proportion as the link between geometry and arithmetic [Benardete,JA]
     Full Idea: The Greeks saw the independent science of proportion as the link between geometry and arithmetic.
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.15)
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
3330 Negatives, rationals, irrationals and imaginaries are all postulated to solve baffling equations [Benardete,JA]
     Full Idea: The Negative numbers are postulated (magic word) to solve x=5-8, Rationals postulated to solve 2x=3, Irrationals for x-squared=2, and Imaginaries for x-squared=-1. (…and Zero for x=5-5) …and x/0 remains eternally open.
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.14)
3337 Natural numbers are seen in terms of either their ordinality (Peano), or cardinality (set theory) [Benardete,JA]
     Full Idea: One approaches the natural numbers in terms of either their ordinality (Peano), or cardinality (set theory).
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.17)
7. Existence / B. Change in Existence / 4. Events / a. Nature of events
3310 If slowness is a property of walking rather than the walker, we must allow that events exist [Benardete,JA]
     Full Idea: Once we conceded that Tom can walk slowly or quickly, and that the slowness and quickness is a property of the walking and not of Tom, we can hardly refrain from quantifying over events (such as 'a walking') in our ontology.
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch. 6)
7. Existence / C. Structure of Existence / 8. Stuff / a. Pure stuff
12793 Early pre-Socratics had a mass-noun ontology, which was replaced by count-nouns [Benardete,JA]
     Full Idea: With their 'mass-noun' ontologies, the early pre-Socratics were blind to plurality ...but the count-noun ontologists came to dominate the field forever after.
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch. 6)
     A reaction: The mass-nouns are such things as earth, air, fire and water. This is a very interesting historical observation (cited by Laycock). Our obsession with identity seems tied to formal logic. There is a whole other worldview waiting out there.
8. Modes of Existence / D. Universals / 6. Platonic Forms / d. Forms critiques
3353 If there is no causal interaction with transcendent Platonic objects, how can you learn about them? [Benardete,JA]
     Full Idea: How can you learn of the existence of transcendent Platonic objects if there is no causal interaction with them?
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.22)
9. Objects / C. Structure of Objects / 5. Composition of an Object
3304 Why should packed-together particles be a thing (Mt Everest), but not scattered ones? [Benardete,JA]
     Full Idea: Why suppose these particles packed together constitute a macro-entity (namely, Mt Everest), whereas those, of equal number, scattered around, fail to add up to anything beyond themselves?
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch. 2)
9. Objects / D. Essence of Objects / 6. Essence as Unifier
3350 Could a horse lose the essential property of being a horse, and yet continue to exist? [Benardete,JA]
     Full Idea: Is being a horse an essential property of a horse? Can we so much as conceive the abstract possibility of a horse's ceasing to be a horse even while continuing to exist?
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.20)
9. Objects / E. Objects over Time / 2. Objects that Change
3309 If a soldier continues to exist after serving as a soldier, does the wind cease to exist after it ceases to blow? [Benardete,JA]
     Full Idea: If a soldier need not cease to exist merely because he ceases to be a soldier, there is room to doubt that the wind ceases to exist when it ceases to be a wind.
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch. 6)
9. Objects / E. Objects over Time / 8. Continuity of Rivers
3351 One can step into the same river twice, but not into the same water [Benardete,JA]
     Full Idea: One can step into the same river twice, but one must not expect to step into the same water.
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.21)
9. Objects / F. Identity among Objects / 5. Self-Identity
3314 Absolutists might accept that to exist is relative, but relative to what? How about relative to itself? [Benardete,JA]
     Full Idea: With the thesis that to be as such is to be relative, the absolutist may be found to concur, but the issue turns on what it might be that a thing is supposed to be relative to. Why not itself?
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch. 8)
3323 Maybe self-identity isn't existence, if Pegasus can be self-identical but non-existent [Benardete,JA]
     Full Idea: 'Existence' can't be glossed as self-identical (critics say) because Pegasus, even while being self-identical, fails to exist.
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.11)
12. Knowledge Sources / A. A Priori Knowledge / 1. Nature of the A Priori
3306 The clearest a priori knowledge is proving non-existence through contradiction [Benardete,JA]
     Full Idea: One proves non-existence (e.g. of round squares) by using logic to derive a contradiction from the concept; it is precisely here, in such proofs, that we find the clearest example of a priori knowledge.
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch. 4)
12. Knowledge Sources / A. A Priori Knowledge / 5. A Priori Synthetic
3349 If we know truths about prime numbers, we seem to have synthetic a priori knowledge of Platonic objects [Benardete,JA]
     Full Idea: Assume that we know to be true propositions of the form 'There are exactly x prime numbers between y and z', and synthetic a priori truths about Platonic objects are delivered to us on a silver platter.
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.18)
3341 Logical positivism amounts to no more than 'there is no synthetic a priori' [Benardete,JA]
     Full Idea: Logical positivism has been concisely summarised as 'there is no synthetic a priori'.
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.18)
3344 Assertions about existence beyond experience can only be a priori synthetic [Benardete,JA]
     Full Idea: No one thinks that the proposition that something exists that transcends all possible experience harbours a logical inconsistency. Its denial cannot therefore be an analytic proposition, so it must be synthetic, though only knowable on a priori grounds.
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.18)
3345 Appeals to intuition seem to imply synthetic a priori knowledge [Benardete,JA]
     Full Idea: Appeals to intuition - no matter how informal - can hardly fail to smack of the synthetic a priori.
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.18)
25. Social Practice / D. Justice / 2. The Law / b. Rule of law
7810 The 'Eumenides' of Aeschylus shows blood feuds replaced by law [Aeschylus, by Grayling]
     Full Idea: The 'Eumenides' of Aeschylus tells how the old rule of revenge and blood feud was replaced by a due process of law before a civil jury.
     From: report of Aeschylus (The Eumenides [c.458 BCE]) by A.C. Grayling - What is Good? Ch.2
     A reaction: Compare Idea 1659, where this revolution is attributed to Protagoras (a little later than Aeschylus). I take the rule of law and of society to be above all the rule of reason, because the aim is calm objectivity instead of emotion.
27. Natural Reality / C. Space / 3. Points in Space
3334 Rationalists see points as fundamental, but empiricists prefer regions [Benardete,JA]
     Full Idea: Rationalists have been happier with an ontology of points, and empiricists with an ontology of regions.
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch.16)
28. God / B. Proving God / 2. Proofs of Reason / a. Ontological Proof
3308 In the ontological argument a full understanding of the concept of God implies a contradiction in 'There is no God' [Benardete,JA]
     Full Idea: In the ontological argument a deep enough understanding of the very concept of God allows one to derive by logic a contradiction from the statement 'There is no God'.
     From: José A. Benardete (Metaphysics: the logical approach [1989], Ch. 4)